where f^{\displaystyle {\hat {f}}} denotes the Fourier transform of f{\displaystyle f}, a(x,ξ){\displaystyle a(x,\xi )} is a standard symbol which is compactly supported in x{\displaystyle x} and Φ{\displaystyle \Phi } is real valued and homogeneous of degree 1{\displaystyle 1} in ξ{\displaystyle \xi }. It is also necessary to require that det(∂2Φ∂xi∂ξj)≠0{\displaystyle \det \left({\frac {\partial ^{2}\Phi }{\partial x_{i}\,\partial \xi _{j}}}\right)\neq 0} on the support of a. Under these conditions, if a is of order zero, it is possible to show that T{\displaystyle T} defines a bounded operator from L2{\displaystyle L^{2}} to L2{\displaystyle L^{2}}.[1]

These need to be interpreted as oscillatory integrals since they do not in general converge. This formally looks like a sum of two Fourier integral operators, however the coefficients in each of the integrals are not smooth at the origin, and so not standard symbols. If we cut out this singularity with a cutoff function, then the so obtained operators still provide solutions to the initial value problem modulo smooth functions. Thus, if we are only interested in the propagation of singularities of the initial data, it is sufficient to consider such operators. In fact, if we allow the sound speed c in the wave equation to vary with position we can still find a Fourier integral operator that provides a solution modulo smooth functions, and Fourier integral operators thus provide a useful tool for studying the propagation of singularities of solutions to variable speed wave equations, and more generally for other hyperbolic equations.